How to Find Exponents That Are Variables


The most surefire way to find exponents that are variables is to use logarithms. Other algebraic methods exist, but they only work for certain types of problems; additionally, such methods tend to take more time. Since logarithms are essentially the inverse of exponentials, it makes sense to use them to find exponential variables. Some background knowledge is required to complete this task; for instance, you must be familiar with terms like base, exponent and variable.

Things You'll Need

  • Calculator that can perform logarithms
  • Pencil
  • Paper
  • Ensure your equation is written such that the base and exponential variable appear on the left side of the equals sign and the single number appears to the right of the equals sign. For instance, rewrite 46 = 5^x so that it reads 5^x = 46. Though this step doesn't affect the solution, it is helpful for explanatory purposes.

  • Take the logarithm of the number on the right side of the equals sign. To do this, press the "log" key on your calculator, followed by the number. Hit the "enter" or "equals" button. Chances are the resultant value is a decimal; if so, round it to the fourth digit. Note this value on your paper. In the case of 5^x = 46, you would find the logarithm of 46, which after rounding is 1.6628.

  • Find the logarithm of the exponent's base, which you've written on the left side of the equals sign. Hit the "log" button on your calculator and then type in the number. Press the "enter" or "equals" key. Again, the resultant value is likely a decimal; if so, round it to the fourth digit. Write this number on your paper. In the previous example, you would take the logarithm of 5, which after rounding produces 0.6989.

  • Divide the result of step 2 by the result of step 3 on your calculator. Essentially, you are dividing the logarithm of the number on the right side of the equation by the logarithm of the base number on the left side of the equation. Round to the digit specified in your individual instructions. In the example, divide 1.6628 by 0.6989. For the sake of consistency, round to the fourth digit, resulting in 2.3792. This is the solution -- the exponential variable, x, equals approximately 2.3792.

  • Check your work by substituting your solution for the exponential variable back into the original equation. Type the base, followed by the "^" symbol and then type the result of step 4. Press the "enter" or "equals" key. If your solution is correct, both sides of the equals sign will reflect approximately the same number. In the case of 5^x = 46, you would perform 5^2.3792, resulting in 46.0245 = 46. Even though the numbers don't match exactly, they differ by only a fractional amount -- due to the previous rounding -- therefore the solution of 2.3792 is correct. If the numbers on each side of the equals sign differ by a significant amount, typically more than 1, then your solution is incorrect; examine your work for errors.


  • Photo Credit Calculator image by Alhazm Salemi from
Promoted By Zergnet


You May Also Like

  • How to Round to the Nearest Tens

    Round the previous digit up one, if the last digit is 5 or higher. Replace the last digit with a 0. If...

  • How to Find Missing Exponents

    Solving for a missing exponent can be as simple as solving 4=2^x, or as complex as finding how much time must pass...

  • How to Determine an Unknown Exponent

    To solve an equation for the exponent, use natural logs in order to solve the equation. Sometimes, you can perform the calculation...

  • How to Round Decimals

    One of the basic rules of arithmetic is that of rounding decimals. Once you have an explanation on how to do so,...

  • How to Simplify Exponents

    Exponents represent shorthand notations of repeated multiplications, often written with the number or variable to be multiplied followed by a superscript value...

Related Searches

Read Article

Can You Take Advantage Of Student Loan Forgiveness?

Is DIY in your DNA? Become part of our maker community.
Submit Your Work!